Questions for the state exam
course "Digital signal processing and signal processors"
(Korneev D.A.)
Correspondence studies
Classification of signals, energy and power of signals. Fourier series. Sine-cosine form, real form, complex form.
CLASSIFICATION OF SIGNALS USED IN RADIO ENGINEERING
From an information point of view, signals can be divided into deterministic And random.
Deterministic call any signal whose instantaneous value at any time can be predicted with probability one. Examples of deterministic signals include pulses or bursts of pulses, the shape, amplitude and time position of which are known, as well as a continuous signal with specified amplitude and phase relationships within its spectrum.
TO random refer to signals whose instantaneous values are unknown in advance and can be predicted only with a certain probability less than one. Such signals are, for example, electrical voltage corresponding to speech, music, a sequence of telegraph code characters when transmitting non-repeating text. Random signals also include a sequence of radio pulses at the input of a radar receiver, when the amplitudes of the pulses and the phases of their high-frequency filling fluctuate due to changes in propagation conditions, target position and some other reasons. You can cite big number other examples of random signals. Essentially, any signal that carries information should be considered random.
The deterministic signals listed above, “fully known,” no longer contain information. In the following, such signals will often be referred to as oscillations.
Along with useful random signals, in theory and practice we have to deal with random interference - noise. The noise level is the main factor limiting the speed of information transmission for a given signal.
Analog signal Discrete signal
Quantized signal Digital signal
Rice. 1.2. Signals arbitrary in magnitude and time (a), arbitrary in magnitude and discrete in time (b), quantized in magnitude and continuous in time (c), quantized in magnitude and discrete in time (d)
Meanwhile, signals from the message source can be either continuous or discrete (digital). In this regard, the signals used in modern radio electronics can be divided into the following classes:
arbitrary in value and continuous in time (Fig. 1.2, a);
arbitrary in value and discrete in time (Fig. 1.2, b);
quantized in magnitude and continuous in time (Fig. 1.2, c);
quantized in magnitude and discrete in time (Fig. 1.2, d).
First class signals (Fig. 1.2, a) are sometimes called analog, since they can be interpreted as electrical models of physical quantities, or continuous, since they are specified along the time axis at an uncountable set of points. Such sets are called continuum. In this case, along the ordinate axis, signals can take on any value within a certain interval. Since these signals may have discontinuities, as in Fig. 1.2, and, then, in order to avoid incorrectness in the description, it is better to designate such signals by the term continuum.
So, the continuous signal s(t) is a function of the continuous variable t, and the discrete signal s(x) is a function of the discrete variable x, which takes only fixed values. Discrete signals can be created directly by the source of information (for example, discrete sensors in control or telemetry systems) or formed as a result of sampling of continuous signals.
In Fig. 1.2, b shows a signal specified at discrete values of time t (at a countable set of points); the magnitude of the signal at these points can take on any value in a certain interval along the ordinate axis (as in Fig. 1.2, a). Thus, the term discrete characterizes not the signal itself, but the way it is specified on the time axis.
Signal in Fig. 1.2, is specified on the entire time axis, but its value can only take discrete values. In such cases, we speak of a signal quantized by level.
In what follows, the term discrete will be used only in relation to time sampling; discreteness in level will be designated by the term quantization.
Quantization is used when representing signals in digital form using digital encoding, since levels can be numbered with numbers with a finite number of digits. Therefore, a signal discrete in time and quantized in level (Fig. 1.2, d) will henceforth be called digital.
Thus, it is possible to distinguish between continuous (Fig. 1.2, a), discrete (Fig. 1.2, b), quantized (Fig. 1.2, c) and digital (Fig. 1.2, d) signals.
Each of these signal classes can be associated with an analog, discrete or digital circuit. The relationship between the type of signal and the type of circuit is shown in the functional diagram (Fig. 1.3).
When processing a continuum signal using an analog circuit, no additional signal conversion is required. When processing a continuum signal using a discrete circuit, two transformations are necessary: sampling the signal in time at the input of the discrete circuit and the inverse transformation, i.e., restoring the continuum structure of the signal at the output of the discrete circuit.
For an arbitrary signal s(t) = a(t)+jb(t), where a(t) and b(t) are real functions, the instantaneous signal power (energy distribution density) is determined by the expression:
w(t) = s(t)s*(t) = a 2 (t)+b 2 (t) = |s(t)| 2.
The signal energy is equal to the integral of the power over the entire interval of the signal's existence. In the limit:
E s = w(t)dt = |s(t)| 2 dt.
Essentially, instantaneous power is the power density of a signal, since power measurements are only possible through the energy released over certain intervals of non-zero length:
w(t) = (1/Dt) |s(t)| 2 dt.
The signal s(t) is studied, as a rule, over a certain interval T (for periodic signals - within one period T), with the average signal power:
W T (t) = (1/T) w(t) dt = (1/T) |s(t)| 2 dt.
The concept of average power can also be extended to continuous signals, the energy of which is infinitely large. In the case of an unlimited interval T, a strictly correct determination of the average signal power is made using the formula:
W s = w(t) dt.
The idea that any periodic function can be represented as a series of harmonically related sines and cosines was proposed by Baron Jean Baptiste Joseph Fourier (1768−1830).
Fourier series function f(x) is represented as
2.1.1.Deterministic and random signals
Deterministic signal is a signal whose instantaneous value at any time can be predicted with a probability equal to one.
An example of a deterministic signal (Fig. 10) can be: sequences of pulses (the shape, amplitude and time position of which are known), continuous signals with given amplitude-phase relationships.
Methods for specifying a MM signal: analytical expression (formula), oscillogram, spectral representation.
An example of a MM of a deterministic signal.
s(t)=S m ·Sin(w 0 t+j 0)
Random signal– a signal, the instantaneous value of which at any time is unknown in advance, but can be predicted with a certain probability, less than unity.
An example of a random signal (Fig. 11) could be a voltage corresponding to human speech or music; sequence of radio pulses at the input of the radar receiver; interference, noise.
2.1.2. Signals used in radio electronics
Continuous in magnitude (level) and continuous in time (continuous or analog) signals– take any values s(t) and exist at any moment in a given time interval (Fig. 12).
Continuous in magnitude and discrete in time signals are specified at discrete time values (on a countable set of points), the magnitude of the signal s(t) at these points takes on any value in a certain interval along the ordinate axis.
The term “discrete” characterizes the method of specifying a signal on the time axis (Fig. 13).
Magnitude-quantized and time-continuous signals are specified on the entire time axis, but the value s(t) can only take discrete (quantized) values (Fig. 14).
Magnitude-quantized and time-discrete (digital) signals– signal level values are transmitted in digital form (Fig. 15).
2.1.3. Pulse signals
Pulse- an oscillation that exists only within a finite period of time. In Fig. 16 and 17 show a video pulse and a radio pulse.
For a trapezoidal video pulse, enter the following parameters:
A – amplitude;
t and – video pulse duration;
t f – front duration;
t cf – cut duration.
S р (t)=S in (t)Sin(w 0 t+j 0)
S in (t) – video pulse – envelope for a radio pulse.
Sin(w 0 t+j 0) – filling the radio pulse.
2.1.4. Special signals
Switching function (single function(Fig. 18) or Heaviside function) describes the process of transition of some physical object from a “zero” to a “unit” state, and this transition occurs instantly.
Delta function (Dirac function) is a pulse whose duration tends to zero, while the height of the pulse increases indefinitely. It is customary to say that the function is concentrated at this point.
 | (2) |
| (3) |
From an information point of view, signals can be divided into deterministic and random.
Deterministic is any signal whose instantaneous value at any time can be predicted with a probability of one. Examples of deterministic signals include pulses or bursts of pulses, the shape, amplitude and time position of which are known, as well as a continuous signal with specified amplitude and phase relationships within its spectrum.
Random signals include signals whose instantaneous values are unknown in advance and can be predicted only with a certain probability less than one. Such signals are, for example, electrical voltage corresponding to speech, music, a sequence of telegraph code characters when transmitting non-repeating text. Random signals also include a sequence of radio pulses at the input of a radar receiver, when the amplitudes of the pulses and the phases of their high-frequency filling fluctuate due to changes in propagation conditions, target position and some other reasons. There are many other examples of random signals that can be given. Essentially, any signal that carries information should be considered random.
The deterministic signals listed above, “fully known,” no longer contain information. In the following, such signals will often be referred to as oscillations.
Along with useful random signals, in theory and practice we have to deal with random interference - noise. The noise level is the main factor limiting the speed of information transmission for a given signal.
Rice. 1.2. Signals arbitrary in magnitude and time (a), arbitrary in magnitude and discrete in time (b), quantized in magnitude and continuous in time (c), quantized in magnitude and discrete in time (d)
Therefore, the study of random signals is inseparable from the study of noise. Useful random signals, as well as noise, are often combined under the term random oscillations or random processes.
A further division of signals can be associated with their nature: we can talk about a signal as a physical process or as numbers encoded, for example in binary code.
In the first case, a signal is understood as any time-varying electrical quantity (voltage, current, charge, etc.) associated in a certain way with the transmitted message.
In the second case, the same message is contained in a sequence of binary-coded numbers.
Signals generated in radio transmitting devices and emitted into space, as well as entering the receiving device, where they undergo amplification and some transformations, are physical processes.
The previous paragraph indicated that modulated oscillations are used to transmit messages over a distance. In this regard, signals in a radio communication channel are often divided into control signals and radio signals; The former are understood as modulating, and the latter as modulated oscillations.
Signal processing in the form of physical processes is carried out using analog electronic circuits (amplifiers, filters, etc.).
The processing of digitally encoded signals is carried out using computer technology.
Shown in Fig. 1.1 and the block diagram of the communication channel described in § 1.2 does not contain instructions about the type of signal used to transmit the message and the structure of individual devices.
Meanwhile, signals from the source of messages, as well as after the detector (Fig. 1.1) can be either continuous or discrete (digital). In this regard, the signals used in modern radio electronics can be divided into the following classes:
arbitrary in value and continuous in time (Fig. 1.2, a);
arbitrary in value and discrete in time (Fig. 1.2, b);
quantized in magnitude and continuous in time (Fig. 1.2, c);
quantized in magnitude and discrete in time (Fig. 1.2, d).
Signals of the first class (Fig. 1.2, a) are sometimes called analog, since they can be interpreted as electrical models of physical quantities, or continuous, since they are specified along the time axis at an uncountable set of points. So? sets are called continuum. In this case, along the ordinate axis, signals can take on any value within a certain interval. Since these signals may have discontinuities, as in Fig. 1.2, and, then, in order to avoid incorrectness in the description, it is better to designate such signals by the term continuum.
So, the continuous signal s(t) is a function of the continuous variable t, and the discrete signal s(x) is a function of the discrete variable x, which takes only fixed values. Discrete signals can be created directly by the source of information (for example, discrete sensors in control or telemetry systems) or formed as a result of sampling of continuous signals.
In Fig. 1.2, b shows a signal specified at discrete values of time t (at a countable set of points); the magnitude of the signal at these points can take on any value in a certain interval along the ordinate axis (as in Fig. 1.2, a). Thus, the term discrete characterizes not the signal itself, but the way it is specified on the time axis.
Signal in Fig. 1.2, is specified on the entire time axis, but its value can only take discrete values. In such cases, we speak of a signal quantized by level.
In what follows, the term discrete will be used only in relation to time sampling; discreteness in level will be designated by the term quantization.
Quantization is used when representing signals in digital form using digital encoding, since levels can be numbered with numbers with a finite number of digits. Therefore, a signal discrete in time and quantized in level (Fig. 1.2, d) will henceforth be called digital.
Thus, it is possible to distinguish between continuous (Fig. 1.2, a), discrete (Fig. 1.2, b), quantized (Fig. 1.2, c) and digital (Fig. 1.2, d) signals.
Each of these signal classes can be associated with an analog, discrete or digital circuit. The relationship between the type of signal and the type of circuit is shown in the functional diagram (Fig. 1.3).
When processing a continuum signal using an analog circuit, no additional signal conversion is required. When processing a continuum signal using a discrete circuit, two transformations are necessary: sampling the signal in time at the input of the discrete circuit and the inverse transformation, i.e., restoring the continuum structure of the signal at the output of the discrete circuit.
Rice. 1.3. Types of signal and corresponding circuits
Finally, when digitally processing a continuous signal, two additional conversions are required: analog-to-digital, i.e., quantization and digital encoding at the input of the digital circuit, and the inverse digital-to-analog conversion, i.e., decoding at the output of the digital circuit.
The signal sampling procedure and especially the analog-to-digital conversion require very high performance of the corresponding electronic devices. These requirements increase with increasing frequency of the continuum signal. Therefore, digital technology has become most widespread when processing signals at relatively low frequencies (audio and video frequencies). However, advances in microelectronics are rapidly increasing the upper limit of processed frequencies.
The term “signal” is often found not only in scientific and technical matters, but also in everyday life. Sometimes, without thinking about the rigor of terminology, we identify concepts such as signal, message, information. This usually does not lead to misunderstandings, since “signal” comes from the Latin term “signum” - “sign”, which has a wide semantic range. Signals are physical means that convey messages. Since electrical signals are the most convenient, their transmission is used in many areas of human activity.
Nevertheless, when starting a systematic study of theoretical radio electronics, it is necessary to clarify, if possible, the substantive meaning of the concept “signal”. In accordance with the accepted tradition, a signal is the process of changing the physical state of an object over time, which serves to display, register and transmit messages.
The range of issues based on the concepts of “message” and “information” is very wide. It is the object of close attention of engineers, mathematicians, linguists, and philosophers.
When starting to study any objects or phenomena, science always strives to carry out their preliminary classification.
Signals can be described using mathematical models. In order to make signals the object of theoretical study and calculations, it is necessary to indicate the method for their mathematical description, i.e. create a mathematical model of the signal under study. A mathematical model of a signal can be, for example, a functional dependence, the argument of which is time.
Creating a model (in this case physical signal) is the first significant step towards a systematic study of the properties of the phenomenon. First of all, the mathematical model allows us to abstract from the specific nature of the signal carrier. In radio engineering, the same mathematical model equally successfully describes current, voltage, electromagnetic field strength, etc.
The essential side of an abstract method based on the concept mathematical model, lies in the fact that we get the opportunity to describe precisely those properties of signals that objectively act as decisively important. In this case, a large number of secondary signs are ignored. For example, in the overwhelming majority of cases it is extremely difficult to select exact functional dependencies that would correspond to the electrical vibrations observed experimentally. Therefore, the researcher, guided by the totality of information available to him, selects from the available arsenal of mathematical models of signals those that are specific situation describe the physical process in the best and simplest way. So, choosing a model is a largely creative process.
Knowing the mathematical models of signals, you can compare these signals with each other, establish their identity and difference, and carry out classification.
From an information point of view, deterministic signals do not contain information, but they can serve as convenient models for studying the temporal and spectral properties of signals.
Real signals containing information appear as random. But mathematical models of such signals are extremely complex and inconvenient for studying the temporal spectral properties of signals.
Deterministic signals are divided into control (low-frequency) and radio signals (high-frequency oscillations). Control signals appear at the place where information occurs (signals various sensors) and can be divided into periodic and non-periodic. This work is devoted to modeling the temporal and spectral properties of periodic signals.
When analyzing periodic signals, their representation using systems of orthogonal functions, for example, Walsh, Chebyshev, Lagger, sine and cosine, and others, has become widespread.
The most widespread is the orthogonal system of basic trigonometric functions - sines and cosines of multiple arguments. This is due to a number of reasons. Firstly, harmonic oscillation is the only function of time that retains its form when passing through any linear circuit(with constant parameters). Only the amplitude and phase of the oscillation changes. Secondly, the decomposition of a complex signal into sines and cosines allows the use of a symbolic method developed for transmission analysis harmonic vibrations through linear circuits. For these, as well as for some other reasons, harmonic analysis has become widespread in all branches of modern science and technology.
If such a signal is presented as a sum of harmonic oscillations with different frequencies, then it is said that spectral decomposition this signal. The individual harmonic components of a signal represent its spectrum. Spectral diagram periodic signal is a graphical representation of the Fourier series coefficients for a specific signal. There are amplitude and phase spectral diagrams, i.e. modules and arguments of complex coefficients of the Fourier series, which completely determine the structure of the frequency spectrum of a periodic oscillation.
They are especially interested in the amplitude diagram, which allows one to judge the percentage of certain harmonics in the spectrum of a periodic signal.
Chapter 1 Elements of the general theory of radio signals
The term “signal” is often found not only in scientific and technical matters, but also in everyday life. Sometimes, without thinking about the rigor of terminology, we identify concepts such as signal, message, information. This usually does not lead to misunderstandings, since the word “signal” comes from the Latin term “signum” - “sign”, which has a wide semantic range.
However, when embarking on a systematic study of theoretical radio engineering, one should, if possible, clarify the substantive meaning of the concept “signal”. In accordance with the accepted tradition, a signal is the process of changing the physical state of an object over time, which serves to display, register and transmit messages. In the practice of human activity, messages are inextricably linked with the information contained in them.
The range of issues based on the concepts of “message” and “information” is very wide. It is the object of close attention of engineers, mathematicians, linguists, and philosophers. In the 40s, K. Shannon completed the initial stage of developing a deep scientific direction - information theory.
It should be said that the problems mentioned here, as a rule, go far beyond the scope of the course “Radio Engineering Circuits and Signals”. Therefore, this book will not outline the relationship that exists between the physical appearance of a signal and the meaning of the message contained in it. Moreover, the question of the value of the information contained in the message and, ultimately, in the signal will not be discussed.
1.1. Classification of radio signals
When starting to study any new objects or phenomena, science always strives to carry out their preliminary classification. Below, such an attempt is made in relation to signals.
The main goal is to develop classification criteria, and also, which is very important for what follows, to establish certain terminology.
Description of signals using mathematical models.
Signals as physical processes can be studied using various instruments and devices - electronic oscilloscopes, voltmeters, receivers. This empirical method has a significant drawback. Phenomena observed by an experimenter always appear as private, isolated manifestations, deprived of the degree of generalization that would make it possible to judge their fundamental properties and predict results under changed conditions.
In order to make signals objects of theoretical study and calculations, one must indicate a method for their mathematical description or, in the language of modern science, create a mathematical model of the signal being studied.
A mathematical model of a signal can be, for example, a functional dependence, the argument of which is time. As a rule, in the future such mathematical models of signals will be denoted by Latin symbols s(t), u(t), f(t), etc.
Creating a model (in this case, a physical signal) is the first significant step towards systematically studying the properties of a phenomenon. First of all, the mathematical model allows us to abstract from the specific nature of the signal carrier. In radio engineering, the same mathematical model equally successfully describes current, voltage, electromagnetic field strength, etc.
An essential aspect of the abstract method, based on the concept of a mathematical model, is that we get the opportunity to describe precisely those properties of signals that objectively act as decisively important. In this case, a large number of secondary signs are ignored. For example, in the overwhelming majority of cases it is extremely difficult to select exact functional dependencies that would correspond to the electrical vibrations observed experimentally. Therefore, the researcher, guided by the totality of information available to him, selects from the available arsenal of mathematical signal models those that in a particular situation best and most simply describe the physical process. So, choosing a model is a largely creative process.
Functions that describe signals can take both real and complex values. Therefore, in the future we will often talk about real and complex signals. The use of one or another principle is a matter of mathematical convenience.
Knowing the mathematical models of signals, you can compare these signals with each other, establish their identity and difference, and carry out classification.
One-dimensional and multidimensional signals.
A typical signal for radio engineering is the voltage at the terminals of a circuit or the current in a branch.
Such a signal, described by a single function of time, is usually called one-dimensional. In this book, one-dimensional signals will be studied most often. However, sometimes it is convenient to introduce multidimensional, or vector, signals of the form
formed by some set of one-dimensional signals. The integer N is called the dimension of such a signal (terminology borrowed from linear algebra).
For example, the system of voltages at the terminals of a multi-terminal network serves as a multidimensional signal.
Note that a multidimensional signal is an ordered collection of one-dimensional signals. Therefore, in the general case, signals with different orders of components are not equal to each other:
Multidimensional signal models are especially useful in cases where the functioning of complex systems is analyzed using a computer.
Deterministic and random signals.
Another principle of classification of radio signals is based on the possibility or impossibility of accurately predicting their instantaneous values at any time.
If the mathematical model of the signal allows such a prediction to be made, then the signal is called deterministic. The ways of setting it can be varied - mathematical formula, a computational algorithm, and finally a verbal description.
Strictly speaking, deterministic signals, as well as deterministic processes corresponding to them, do not exist. The inevitable interaction of the system with the physical objects surrounding it, the presence of chaotic thermal fluctuations and simply incomplete knowledge about the initial state of the system - all this forces us to consider real signals as random functions of time.
In radio engineering, random signals often manifest themselves as interference, preventing the extraction of information from a received vibration. The problem of combating interference and increasing the noise immunity of radio reception is one of the central problems of radio engineering.
The concept of a "random signal" may seem contradictory. However, it is not. For example, the signal at the output of a radio telescope receiver aimed at a source of cosmic radiation represents chaotic oscillations, which, however, carry a variety of information about a natural object.
There is no insurmountable boundary between deterministic and random signals.
Very often, in conditions where the level of interference is significantly less than the level of a useful signal with a known shape, a simpler deterministic model turns out to be quite adequate to the task.
The methods of statistical radio engineering, developed in recent decades to analyze the properties of random signals, have many specific features and are based on the mathematical apparatus of probability theory and the theory of random processes. A number of chapters of this book will be entirely devoted to this range of issues.
Pulse signals.
A very important class of signals for radio engineering are pulses, i.e. oscillations that exist only within a finite period of time. In this case, a distinction is made between video pulses (Fig. 1.1, a) and radio pulses (Fig. 1.1, b). The difference between these two main types of impulses is as follows. If is a video pulse, then the corresponding radio pulse (frequency and initial are arbitrary). In this case, the function is called the envelope of the radio pulse, and the function is called its filling.
Rice. 1.1. Pulse signals and their characteristics: a - video pulse, b - radio pulse; c - determination of the numerical parameters of the pulse
In technical calculations, instead of a full mathematical model that takes into account the details of the “fine structure” of the pulse, numerical parameters are often used that give a simplified idea of its shape. Thus, for a video pulse close in shape to a trapezoid (Fig. 1.1, c), it is customary to determine its amplitude (height) A. From the time parameters, the duration of the pulse is indicated, the duration of the front and the duration of the cutoff
In radio engineering, we deal with voltage pulses whose amplitudes range from fractions of a microvolt to several kilovolts, and whose durations reach fractions of a nanosecond.
Analog, discrete and digital signals.
Finishing short review principles of classification of radio signals, we note the following. Often the physical process that produces a signal evolves over time in such a way that the signal values can be measured in. any moment in time. Signals of this class are usually called analog (continuous).
The term “analog signal” emphasizes that this signal is “analogous”, completely similar to the physical process that generates it.
A one-dimensional analog signal is clearly represented by its graph (oscillogram), which can be either continuous or with break points.
Initially, radio engineering used exclusively analog signals. Such signals made it possible to successfully solve relatively simple technical problems (radio communications, television, etc.). Analogue signals were easy to generate, receive and process using the means available at the time.
Increased demands on radio systems and a variety of applications have forced us to look for new principles for their construction. In some cases, analog ones have been replaced by pulsed systems, the operation of which is based on the use of discrete signals. The simplest mathematical model of a discrete signal is a countable set of points - an integer) on the time axis, in each of which the reference value of the signal is determined. Typically, the sampling step for each signal is constant.
One of the advantages of discrete signals over analog signals is that there is no need to reproduce the signal continuously at all times. Due to this, it becomes possible to transmit messages from different sources over the same radio link, organizing multi-channel communication with time-separated channels.
Intuitively, fast time-varying analog signals require a small step size to be sampled. In ch. 5 this fundamentally important issue will be explored in detail.
A special type of discrete signals are digital signals. They are characterized by the fact that the reference values are presented in the form of numbers. For reasons of technical convenience of implementation and processing, binary numbers with a limited and, as a rule, not too large number of digits are usually used. Recently, there has been a trend towards widespread implementation of systems with digital signals. This is due to the significant advances achieved by microelectronics and integrated circuit technology.
It should be borne in mind that in essence any discrete or digital signal(we are talking about a signal - a physical process, and not about a mathematical model) is an analog signal. Thus, an analog signal slowly changing over time can be associated with its discrete image, which has the form of a sequence of rectangular video pulses of the same duration (Fig. 1.2, a); the height of ethnh pulses is proportional to the values at the reference points. However, you can do it differently, keeping the height of the pulses constant, but changing their duration in accordance with the current reference values (Fig. 1.2, b).
Rice. 1.2. Sampling of an analog signal: a - with variable amplitude; b - with variable duration of counting pulses
Both analog signal sampling methods presented here become equivalent if we assume that the values of the analog signal at the sampling points are proportional to the area of the individual video pulses.
Recording of reference values in the form of numbers is also carried out by displaying the latter in the form of a sequence of video pulses. The binary number system is ideally suited for this procedure. You can, for example, associate a high potential level with one and a low potential level with zero, f Discrete signals and their properties will be studied in detail in Chapter. 15.
